Vibrations of Strings: Sonometer and Harmonics Explained

Stretch a string between two fixed ends and pluck it, and it doesn't vibrate randomly — it settles into standing-wave patterns with still points (nodes) and wildly swinging points (antinodes). Only certain wavelengths "fit", giving the harmonics: the fundamental plus its whole-number multiples. This is the physics of every guitar, violin, piano, and sitar string.

The school sonometer — a stretched wire over a sounding box with movable bridges — verifies the three laws of vibrating strings and measures unknown frequencies by comparison. The same laws explain frets, tuning pegs, and string gauges.

Key Concepts

• Standing wave: two identical travelling waves moving in opposite directions superpose into a pattern that oscillates in place instead of travelling.
• Node: point of zero motion; antinode: point of maximum motion. Fixed ends are always nodes.
• Fundamental (1st harmonic): half a wavelength fits: λ_1 = 2L, f_1 = v/(2L).
• Harmonics: f_n = n f_1 for n = 1, 2, 3, … — the 2nd harmonic (octave) has 2 half-wavelengths, the 3rd has 3, and so on. A plucked string sounds all of them at once; their mix is the timbre.
• Sonometer: lab instrument with a stretched wire, two movable bridges setting the vibrating length, and weights setting the tension.

Formulas

• f_n = nv/(2L), n = 1, 2, 3, …, with v = √(T/μ). Units: f in Hz, L in m, T in N, μ in kg/m.
• Laws of vibrating strings (what the sonometer verifies): f ∛ 1/L (law of length), f ∛ √T (law of tension), f ∛ 1/√μ (law of mass).
• Assumptions and limits: perfectly flexible uniform string, fixed ends, small amplitudes. Real strings have slight stiffness, making high harmonics a touch sharp (inharmonicity — piano tuners stretch octaves to compensate).

Worked Example

A sonometer wire: L = 0.60 m, T = 90 N, μ = 0.004 kg/m.

1. Wave speed: v = √(T/μ) = √(90/0.004) = √22500 = 150 m/s.
2. Fundamental: f_1 = v/(2L) = 150/(2 x 0.60) = 150/1.2 = 125 Hz.
3. Harmonics: f_2 = 250 Hz, f_3 = 375 Hz, f_4 = 500 Hz — exact whole-number multiples.
4. Law of length: halve the length (L = 0.30 m): f_1 = 150/0.60 = 250 Hz — doubled, as 1/L predicts. This is the guitarist's 12th fret (one octave up).
5. Measuring a fork: adjust L until the wire resonates with the fork; then f_fork = v/(2L).

Common Misconceptions

• "Touching the string's midpoint silences it." It kills the fundamental and all even… actually it kills modes with an antinode at the midpoint, isolating the 2nd harmonic — the glassy "harmonic" (flageolet) guitarists play. The string still sings, one octave up.
• "Harmonics are always exact multiples." True for ideal strings, but real stiff strings run slightly sharp at high n. Piano tuners deliberately stretch octaves so the sharp harmonics of low notes match the fundamentals of high notes.
• "The sonometer measures tension directly." It compares frequencies: you find unison between wire and tuning fork, then compute whatever is unknown (T, μ, or f) from f = v/(2L).

Real-World Applications

• Guitars and violins: frets/fingers set L, pegs set T, string gauge sets μ — the three laws as a musical instrument.
• Pianos: 230-odd strings sized by these laws; bass strings are wound (more μ) to go low without absurd length.
• School labs: sonometers verify the three laws and measure unknown tuning-fork frequencies.
• Non-destructive testing: the resonant frequency of a tensioned cable reveals its tension — used on bridges and cranes.

In short: a fixed string vibrates only at f_n = nv/(2L), with f ∛ 1/L, √T, 1/√μ — the three laws the sonometer verifies. Frets shorten, pegs tighten, gauges weighten: every stringed instrument is these laws made musical.